Ribbon Paths and Jack Polynomials

We introduce a generalization of Motzkin paths that we call “ribbon paths.” For a particular choice of weights, we express weighted sums of ribbon paths in terms of Jack polynomials. As a consequence, in light of the work of Chapuy-Dolega, our weighted sums are also equal to weighted sums of certain ribbon graphs on non-oriented real surfaces. Our results are new even for ribbon paths without “slides,” in which case the weighted sums are expressed through Schur polynomials and the corresponding real surfaces are oriented.